Having experimented with theremins for many years now, and gone
through the whole gamut of empirical approaches to theremin design, I
thought it was time to do a theoretical analysis on that elusive quality
referred to as theremin "sensitivity."

By sensitivity I mean essentially: how much of a change in pitch
results from a given change in hand position? The same criterion could
of course be applied to the volume controller of the theremin, but in this
discussion I will refer only to the pitch aspect.

Sensitivity can be compared with to "scale" of a string instrument. A
bass viol, for instance, would have a very low sensitivity compared to a
1/4-size violin. In other words, all other things being equal, a
semitone change on a bass requires a much greater hand movement than
would a semitone change on a violin.

Similarly, there have been many theremin designs over the years, with
widely varying degrees of sensitivity. To start our analysis of some of
these designs, we have to recognize that it is the change in
capacitance between the hand and the pitch antenna that
causes a change in pitch. We also have to define some sort of standard
of measurement. I propose that sensitivity be defined in kilohertz
per picofarad. In other words, by how many kilohertz will the pitch
change for a one picofarad change in hand-to-antenna capacitance. This
results in easily manageable numbers, that range (depending on design)
from less than 0.2 to well over 3.0.

2) Analysing Sensitivity:

Let's start with the basics. All theremin designs (well, okay; almost
all) use the LC resonant circuit (L = inductance, as of a coil; C =
capacitance) as the frequency-defining elements. Such a circuit is at
the heart of the oscillators used in most theremins; the rest of the
circuitry provides the positive feedback required to sustain
oscillation.

Such LC circuits resonate
at the frequency at which the inductive reactance of the coil [] equals the capacitive
reactance of the capacitor []. Setting these two equations equal to each other and
solving for F gives the well-known resonance formula:

Or, combining constants and scaling for direct use of microhenries
and picofarads, we get:

Frequency:

where F is in kilohertz, L is in microhenries (µH), and C is in picofarads (pF).

It should be noted that the value for C is the total value of all
elements in parallel, including fixed and variable "tank" capacitors,
stray circuit capacitances, parasitic antenna capacitance, and of course
the all-important hand-to-antenna capacitance. The largest component
will usually be the "tank" capacitances, consisting usually of a fixed
capacitor (on the order of 100 to 1000 pF), often in parallel with a
variable capacitor used for tuning. The parasitic antenna capacitance
will usually be on the order of 20 to 50 pF, depending on the geometry
and size of the antenna. Stray circuit capacitances are usually minimal,
on the order of a few picofarads. The additional hand-to-antenna
capacitance is usually the smallest component of the total capacitance,
contributing from zero (at an "infinite" distance from the antenna) to
only a few picofarads very close to the antenna.

Incidentally, the resonance formula can be used to "back-calculate"
the inductance in vintage theremin designs, knowing the tank capacitance
and frequency of oscillation. Re-arranging the formula above, we
get:

(L in µH,
F in kHz, C in pF)

For instance, the hand-drawn schematic of "Clara Rockmore's Theremin"
shows a 620 pF fixed capacitor in parallel with a 43 pF variable cap and
another little unmarked trim cap in the pitch oscillator tank. We can
estimate parasitic antenna capacitance at about 50 pF, since she was
known to use large rod-type antennas. The grid capacitance of the 27
tube was on the order of 10 pF, so we can estimate that the total tank
capacitance was probably on the order of 750 pF. Unfortunately, the
circuit gives no indication of inductance, or even of frequency of
oscillation.

However, we can take a clue from the liner notes of "The Art of the
Theremin," in which it is stated that the oscillating frequency was on
the order of a surprisingly low 170 kHz. Plugging these values into our
formula for L, we get

= 1170
µH (or 1.17 mH)

The same technique was used to derive an approximate value for the
coils used in Doug Forbes' tube design, given his report that the
oscillator runs at about 700 kHz. with the capacitor plates at
half-mesh.

Back to the issue of sensitivity. Another way of writing our
resonance formula is:

Taking the partial derivative with respect to C gives us a direct
expression of theremin sensitivity, in kilohertz per picofarad:

Sensitivity:

Since we took a partial derivative on C, assuming L to be constant,
the quantity in the brackets can be lumped together as a constant. Note that the
minus sign indicates that frequency decreases with increasing
capacitance.

To put the result into words, sensitivity is inversely proportional to the
square root of tank capacitance cubed.

Similarly, we can take a partial relative to L, and achieve a similar
result; we only have to interchange L and C in the above
relationships.

3) Results:

The easiest way to play with this is using a spreadsheet such as
Lotus 1-2-3, Excell, or the spreadsheet option in any version of
Microsoft Works. You can enter colums for L and C for inputting
constants, and columns for F and dF/dC for inputting the appropriate
formulas. You can then experiment to your heart's content with various
values for L and C, getting immediate reports of center-frequency (F)
and sensitivity (dF/dC). You can download the worksheet in
1-2-3 format or in
Excel format
by right-clicking on the link and selecting
"Save Link As..." or "Save Target As...".

The results of such plug 'n' play experimentation using a few known theremin designs
is also shown below:

Design description

Induct.L

uH

Tank C

pF

Center F

kHz

dF/dC

kHz/pF

Clara Rockmore

1165

750

170.266

-0.114

SWTP ("TECI") max L

300

410

453.803

-0.553

SWTP ("TECI") min L

150

410

641.775

-0.783

Theremax max L

350

120

776.597

-3.236

Theremax min L

180

120

1082.913

-4.512

Doug Forbes C=150

500

150

581.446

-1.938

Doug Forbes C=100

500

100

711.763

-3.559

"Theoretical" 1

200

200

796.178

-1.990

"Theoretical" 2

100

400

796.178

-0.995

As you can see, there is a wide range of sensitivity from the lowest (Clara
Rockmore) to the highest (Theremax at minimum inductance). The two
designs with adjustable inductors (coils) are shown for both extremes of
inductance. This demonstrates the dependance on inductance, as predicted
by the partial derivative dF/dL. On such designs, you can experiment
with different sensitivities, depending on where you set the coils.
Similarly, Doug Forbes' design allows a large range of adjustment on the
tank capacitor, so sensitivity is shown at two settings (150 pF and 100
pF total).

Two "theoretical" entries are shown also. Both have the same center
frequency, but use different ratios of L and C. As you can see,
increasing C (while decreasing L to maintain the same resonant
frequency) causes a decrease in sensitivity dF/dC.

The general conclusions are that increasing the center frequency (by
decreasing inductance or capacitance or both) increases sensitivity.
Furthermore, for a given frequency, we can increase sensitivity by
maximizing L and minimizing C.

This of course only addresses the electronic aspect of sensitivity.
The mechanical geometry of the antenna and even the size of the
player's hand can have some effect. However, experience has shown that
these effects are small compared with the electronic aspects.

So, more sensitivity is better, right? Not necessarily. It is
interesting to note that the original theremin designs had a very low
sensitivity compared to the more modern approaches. This means that
comparatively large hand motions were required to effect a given pitch
change. On the other hand, it means that such older designs are much
less "touchy", and therefore a lot easier to play. In my opinion we're
making an already difficult instrument even more difficult, even
impossible, by increasing sensitivity to the point of absurdity.

4) Variance of Capacitance:

I'll finish with a brief discussion on hand capacitance. The
relationship between distance from the antenna and capacitance
approximately follows an inverse-square-law. This means that as one
approaches the antenna at a constant rate, the capacitance increases
very slowly at first, then increases faster and faster as the hand gets
closer to the antenna. Very close to the antenna, the capacitance
increases astronomically, making the range within an inch or so from the
antenna virtually unusable except for "special effects."

This effect is partially offset by the nature of the LC tuned
circuit, the frequency of which depends on the inverse square-root of
capacitance. This is the main reason, I believe, why oscillators based
on a single pole (only one reactive component, i.e. capacitance) never
took hold for theremin use. I, and probably many others, have
experimented with RC oscillators in an effort to get rid of those pesky
coils; even the ordinary NE555 timer can be used for this purpose.
However, in such circuits the oscillating frequency is inversely
proportional to capacitance, rather than square-root of capacitance, and
the "square-law" effect is correspondingly a lot worse. Another way of
looking at this is that sensitivity (dF/dC) of the RC circuits is
proportional to
1/C

2 instead of
1/C1.5 in the case of
the LC circuit.

Lest anyone think that this means we're stuck with coils, don't
despair. I'm presently working up a coil-less design that behaves
exactly like the LC circuits traditionally used. The trick is to use a
"virtual inductor" using a high-speed op-amp to simulate a coil.
Similarly, varactor diodes can be used instead of mechanical trimmer
capacitors to put a modern twist on an old idea, and allow for greater
flexibility and improved stability.

As a final note, choice of sensitivity depends a great deal on your
playing style. If you want more accurate pitch control, especially at
higher frequencies, choose a design with a low sensitivity. If you work
mainly with lower frequencies, use a higher sensitivity; you won't need
as much motion to effect a given pitch change.